Bayesian Analysis of Hemispherical Power Asymmetry in the Cosmic Microwave Background

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Bayesian Analysis of Hemispherical Power Asymmetry in the Cosmic Microwave

Background

Candidatus Scientiarum Thesis by Jostein Hoftuft

Institute of Theoretical Astrophysics University of Oslo

June, 2009

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This work, entitled “Bayesian Analysis of Hemispherical Power Asymmetry in the Cos- mic Microwave Background” is distributed under the terms of the Public Library of Sci-

ence Open Access License, a copy of which can be found at http://www.publiclibraryofscience.org.

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Preface

The current cosmological standard model is based on inflation. This theory predicts that the initial perturbation in the universe had a Gaussian distribution on any given scale, and we would expect that the universe is statistically isotropic and homogeneous, in accordance with the cosmological principle.

The cosmic microwave background (CMB) radiation is basically a snapshot of the universe when it was around 400 000 years old. This radiation has small fluctuations in its temperature as imprinted by the density perturbations at the time it was sent out. After that time, the change in the radiation has been linear, so the characteristics of the radiation still remain the same today. By measuring this radiation today, we are basically probing the earliest times of the universe. From this radiation we can extract information about the universe, such as its age, contents and geometry. Several experiments have set out to detect this ancient radiation. COBE was the first experiment to find the small perturbations in the CMB radiation. Using a beam with FWHM at 7^◦, the resolution of the observations was low. Later, experiments like BOOMERanG observed the CMB anisotropies at a considerably higher resolution, but only measuring a small patch of the sky. This gave the rise to the standard cosmological model.

WMAP performed a high resolution measurement of the entire sky and confirmed the cosmological model and the cosmological parameters could now be determined with greater accuracy. The predicted theory agrees well with the observations of the CMB, and we should therefore expect the universe to be isotropic and homogeneous. But the full-sky maps from WMAP also showed some unexpected features. Several studies of the maps have revealed strong hints of non-Gaussianity and violation of statistical isotropy.

One of these properties was an observed asymmetry in the map. On large scales there were large fluctuations in temperature in one hemisphere, while the other did not have as large fluctuations on the same scales. This asymmetric distribution of power on opposing hemispheres seems to be a violation of isotropy. Several independent works have pointed out this feature. The fluctuation power for large scales seems to have a dipolar distribution.

To try and characterize this distribution, I adopt a simple model where the isotropic CMB field is modulated by a dipole field. The analysis is done using a Bayesian framework. The purpose is to find the strength (amplitude) and direction of this field, and for how small scales this asymmetry is present. The method used here to find these parameters is a time consuming process, as it demands an inversion of a large

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scales we could compute using this method.

The analysis was done using the five-year WMAP data. The results shows evidence for this model at the 3.5σ level, and an article on the results is to be published in Astrophysical Journal.

Acknowledgments

I would like to give a big thanks to Hans Kristian Eriksen, who has been my supervisor.

He has been a source of inspiration through out the whole period, and has always been helpful, conveyed his knowledge in a understandable way and given me much positive feedback. I would also like to thank all the people in Astrokjelleren who have made my time here more fun and exiting, and everybody on Uglelaget.

The computation of this thesis were done using Titan, a computer cluster owned and maintained by the University in Oslo and NOTUR. Throughout the thesis I have used HEALPix.

Jostein Hoftuft

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List of Figures

1.1 Blackbody spectrum of the CMB . . . 16

1.2 CMB anisotropy map by COBE . . . 17

3.1 Decomposition of a temperature map into spherical harmonics . . . 31

3.2 Angular power spectrum . . . 33

3.3 Gaussian beams . . . 36

3.4 Noise estimates . . . 39

3.5 Power spectra for CMB signal and noise . . . 39

4.1 Nested likelihood contours . . . 45

4.2 Convergence for MCMC chains . . . 48

5.1 The quadrupole and octopole alignment . . . 52

5.2 WMAPs ILC sky map and observable asymmetry on large scales . . . . 52

5.3 Simulated map with modulation field . . . 57

5.4 Simulated maps with and without modulation field . . . 59

6.1 Proposal for the modulation axis ˆp . . . 62

6.2 Time per sample for program . . . 68

6.3 Posterior distribution of amplitude and axis of modulation field . . . 69

7.1 Contributions from CMB and foreground radiation . . . 74

7.2 WMAPs V-band temperature map . . . 76

7.3 KQ85 mask . . . 77

7.4 Downgraded masks . . . 79

7.5 Foreground template . . . 80

8.1 Posterior distribution of amplitude and axis of modulation field . . . 84

A.1 Hierarchical partition . . . 104

A.2 Healpix resolutions . . . 105

B.1 Block-cyclic distribution . . . 108

B.2 Matrix partitioning example . . . 110

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Part I

Introducing the CMB and its

properties

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Chapter 1

Introduction

1.1 Introduction

For centuries mankind has been searching for how the universe was created. Almost every culture has a story of how the world was created. Most of these myths are tremendous tales of battles of Gods and other supernatural happenings. The first to explain how the the universe originated using physical laws and scientific description were the Greeks. The word cosmology also came from Greek, k´osmos meaning world;

emphl´ogos meaning knowledge or science. But it is only in the present century that we have developed good theories that describe the universe as a whole, and have observations that support these theories. The modern picture of the universe is that it started with a hot Big Bang, and one observation that strengthen this view of the universe is the presence of a low energy radiation which fills the whole universe. This radiation stems from the Big Bang and is called the cosmic microwave background. In this chapter I will look at how it was discovered and how it changed our view of the universe.

1.2 The discovery of the CMB

In 1915 Albert Einstein submitted his paper on the general theory of relativity, where the field equations describe gravity as a curved spacetime caused by matter and energy [1]. In 1922 Alexander Friedmann found a solution to the equations in which the universe may expand or contract. At the time, Einstein believed that the universe was static, and he introduced a new term with a cosmological constant to the field equations for them to allow a static universe. But a static universe was unstable in this theory, and the universe would start to expand or contract if it was slightly brought out of equilibrium. Later, in 1927, George Lemaˆıtre also published a report where he presented the idea of an expanding universe [2]. Here he also derived the connection between distance and redshift, later to be known as the Hubble law.

By studying the redshift of distant galaxies, Edwin Hubble found a linear relation between the distance and velocity of galaxies. He released this in a paper in 1929,

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which strongly supports an expanding universe [3]. It is then easy to imagine that the universe was smaller before, and that it also was much denser and more hot.

Lemaˆıtre later proposed that the universe expanded from an initial point, called the “Primeval Atom”. This idea was published in Nature in 1931 [4]. He described his theory as “the Cosmic Egg exploding at the moment of the creation”.

But not everyone agreed with this interpretation of the expanding universe. This creation seemed philosophically troubling, as this imply a cause or a creator. There- fore, many scientist, e.g., Fred Hoyle, Thomas Gold and Hermann Bondi, argued for a steady-state universe. To explain that the universe is expanding while still obeying the cosmological principle, matter must be continuously created as it expands in order for the average density to remain at the same level at any given time. That way the universe looks the same at all times. This universe model has no beginning and no end, and therefore, the universe is infinitely old.

Fred Hoyle, who was against this expanding universe model, would later give this model the now famous name Big Bang. In a BBC radio broadcast in 1949 he said,

“This big bang idea seemed to me to be unsatisfactory even before examination show that it leads to serious difficulties. For when we look at our own galaxy there is not the smallest sign that such an explosion ever occurred.” However, there were some who predicted that there would be a remnant signal filling the universe after the Big Bang.

In 1946 George Gamow was pondering the cosmic abundances of the elements. He realized that a newborn, dense universe must be hot enough for the nuclear reactions that create the elements to occur. Two years later Gamow and Ralph Alpher published a paper called “The Origin of Chemical Elements”. Later, detailed calculations by Alpher and Ralph Herman showed that Gamow’s idea did not produce elements heavier than helium. However, in 1957 Fred Hoyle released a paper about stellar nucleosynthesis which explained the formation of heavier elements. Gamow’s theory of the Big Bang nucleosynthesis explained that the early universe must have been very dense and hot, and where radiation and matter was in thermal equilibrium. Under these conditions the photons should have a blackbody spectrum. In 1948 Alpher and Herman published a paper on how this blackbody radiation would cool off as the universe expanded, and they predicted that the universe today would be filled with a blackbody radiation with a temperature of 5 K. But astronomers and scientist did not make any effort to detect this leftover blackbody radiation from the early universe, much due to the lack of interest and the immaturity of microwave observations.

In the sixties Bell Labs built a giant antenna in Holmdel, New Jersey, for long distance radio communication. Two employees of Bell Labs, Arno Penzias and Robert Wilson, saw the potential to use this as a radio telescope. When a satellite took over the job of the antenna, it could now be used for research. When Penzias and Wilson started to use it, they registered a faint signal, or background noise, in the microwave range that remained no matter where they turned their antenna on the sky. Numerous attempts were done to find the source of the signal. They ruled out urban interference and radiation from our own galaxy. The signal did not change with the seasons either.

They even removed the pigeons living in the antenna and scrubbed it clean, but the signal remained. Penzias and Wilson were aware that a 3 K blackbody would produce

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1.3 The search for CMB anisotropies 15

their interference, but unaware of what might be the source of this.

At the same time and just miles away, Robert Dicke at Princeton University and his postdoctoral student, P.J.E. Peebles, had been calculating the blackbody radiation left over from the Big Bang. Peebles calculated that the radiation would have a temperature of 10 K. They were interested in searching for this remnant of the early universe. When a friend told Penzias about the calculation of the radiation left over from the Big Bang by Peebles, he started to realize the significance of their discovery. Penzias called Dicke and in 1965 the pieces of the puzzle fell together. Penzias and Wilson had detected the 2.73 K blackbody radiation that fills the Universe. This relic of the Big Bang is called the Cosmic Microwave Background(CMB), and is uniform over the full sky, i.e., the temperature is the same in any direction. Dicke, Peebles, and their co-workers at Princeton wrote an article for the Astrophysical Journal Letters where they explain that the CMB strongly supports the Big Bang theory [5], while Penzias and Wilson wrote an accompanying letter where they confirmed the existence of the background noise [6]. Penzias and Wilson received the Nobel Prize for Physics for their discovery in 1978.

1.3 The search for CMB anisotropies

The measurements from Penzias and Wilson showed an approximate isotropy in the CMB temperature, just as one would expect if the radiation is produced by the Big Bang. But there are many physical effects that can cause inhomogeneities in the CMB.

Inhomogeneities in the density or velocity of matter in the Universe would cause fluctuations or anisotropies in the CMB. At last-scattering, gravitational instability theory predicts that fractional density perturbations must have been δ & 10⁻³ in order for galaxies and clusters to develop to what we observe today. A challenge in cosmology has been to detect a corresponding fluctuation in the temperature of the CMB radiation. During the sixties and seventies, experimental and theoretical work put harder constraints on the observable fluctuations in the CMB. Sunyaev predicted in 1977 that fluctuations in the CMB must exist a level ∆T /T₀ ∼ 10⁻⁵. New experiments with higher accuracy had to be conducted. The earth’s atmosphere consists of water vapor and it absorbs much of the microwave radiation. To reduce the effects of atmospheric disturbance in order to get a better signal from the CMB, new experiments were performed from balloons or in space.

1.3.1 COBE - the first fluctuations

The Cosmic Background Explorer (COBE) satellite was launched in 1989. The FIRAS instrument on board showed a perfect fit of the CMB and the theoretical curve for a black body at a temperature of 2.7K. See Figure 1.1. Despite the measurement of the CMB by Penzias and Wilson, not everybody embraced the Big Bang theory as the explanation of this. They claimed other sources could explain the radiation. But this new measurement that shows a remarkable agreement between the prediction of Big Bang theory and observations should be enough to convince most skeptics. On

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Figure 1.1: Intensity of the cosmic microwave background as a function of wavelengthas measured by the Far InfraRed Absolute Spectrophotometer (FIRAS). The full line shows the theoretical blackbody curve. Inside of this curve are dozens of measured points, and the error bars are so small that they are obscured by the thickness of the curve. Courtesy of the COBE science team.

board the satellite was also the Differential Microwave Radiometer (DMR) whose job was to measure the anisotropy in the CMB. This is very difficult to measure since it is only deviates from the average temperature as one part in 100 000. The instrument measured the temperature with a beam covering 7^◦ of the sky. The anisotropies are showed in Figure 1.2. The two principal investigators on the DMR and FIRAS, George Smoot and John Mather, respectively, won the Nobel Price in Physics in 2006 for their work. The Nobel Price committee stated that “These measurements also marked the inception of cosmology as a precise science.”

The next step was now to measure the anisotropies on smaller angels. The Toco experiment was the first to localize the first acoustic peak in the power spectrum [7].

1.3.2 BOOMERanG - a flat universe

The BOOMERanG experiment (Balloon Observations of Millimetric Extragalactic Radiationand Geophysics) was a balloon based experiment to measure the the CMB.

By sending a probe 42,000 meters above mean sea level, it was possible to reduce the atmospheric absorption of microwaves and also save a lot of money compared to a satellite mission. The probe was only able to scan a small part of the sky, but it

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1.3 The search for CMB anisotropies 17

Figure 1.2: These pictures shows the anisotropy in the CMB as measured by DMR.

Top: A map including the dipole and galaxy. Due to the earths movement through space relative to the CMB, the frequency of the radiation will be shifted. Moving away relative to the radiation causes a drop in temperature (redshift), while moving towards it causes a rise (blueshift). Middle: The dipole is removed. The band that stretches across the picture is radiation caused by our own Galaxy. Bottom: Galactic foreground emission is reduced and shows us the anisotropies in the CMB. Courtesy of the COBE science team.

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captured large, high fidelity images of the CMB temperature anisotropies. Together with other experiments it could determine the angular diameter distance of the surface of last scattering with high precision, and it determined the geometry of the Universe to be flat. For the 2003 flight it gathered extremely high signal-to-noise ratio maps of the CMB temperature anisotropy, and a measurement of the polarization of the CMB.

1.3.3 WMAP - concordance model with slight flaws

In 2001 NASA launched theWilkinson Microwave AnisotropyProbe (WMAP). Its mission was to measure the temperature difference in the CMB radiation. These anisotropies can then be used to measure the universes geometry, contents and evolution.

To do so, the instrument creates a full-sky map with a 13 arcminutes resolution of the temperature anisotropy. Compared to COBE with a resolution of 7^◦ full-sky map, this had 33 times better resolution. WMAP was the first to localize the third peak.

The data from WMAP gave rise to a standard cosmological model: A flat universe composed of matter, baryons and dark energy with a nearly scale-invariant spectrum of primordial fluctuations. Together with other cosmological data, values of the different cosmological parameters could now be calculated. The age of the Universe is 13.72± 0.12 Gyr. It consists of 4.56±0.15 % baryonic matter, 22.8±1.3 % dark matter and 72.6±1.5 % dark energy [8].

But even if the data described the universe very well, there were some strange features in the map from WMAP. (1): alignments and symmetry features among low low-l multipoles (Tegmark et al. 2003 [9]; de Oliveira-Costa et al. 2004 [10]; Eriksen et al. 2004 [11]) (2): an apparent asymmetry in the distribution of fluctuation power in two opposing hemispheres (Eriksen et al. 2004 [12]; Hansen et al. 2004 [13]) (3): a peculiar cold spot in the southern hemisphere (Vielva et al. 2004 [14]; Cruz et al. 2005 [15])

The reason for this asymmetry is not known, but three possible candidates that can cause this are systematics, foregrounds or some new, exotic physics.

1.3.4 Future missions

New experiments are planned to get better data about the cosmic microwave background radiation. The Planck satellite was launched on 14th of May 2009. This will be the third space CMB mission after COBE and WMAP. The goals of Planck are to make maps for the temperature and polarization anisotropies of the CMB with high resolution (down to 5’) over the entire sky. It will take measurements in a wide range of frequencies (nine frequencies between 30 - 857 GHz) to discriminate between Galactic emission and cosmic signal and to study galaxy clusters and extragalactic point sources.

By measuring the CMB on smaller scales than WMAP, it will be able to extract more information about the cosmological parameters.

It will give us a higher resolution of the CMB over the whole sky. It will detect both the intensity and polarization of photons at nine different frequencies.

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Chapter 2

The physics of the CMB

In this section I will be looking at the physics in the early universe and explain how the CMB radiation originated and later evolved with the universe. I will mostly just give a short introduction to the various physics and not introduce a lot of mathematics. The main target in my thesis is not to explain how the universe started, but how to compute cosmological parameters. But it is nice to have a feeling of what we are studying. For more complete texts on this matter I refer the interested reader to Dodelson [16] and Tegmark [17]. Some of the material in this section is taken from the lecture notes for the course AST4220 by Øystein Elgarøy [18].

2.1 Inflation

The Big Bang explains many features of our universe such as the origin of light elements (Big Bang nucleosynthesis), the formation of the cosmic microwave background, and the relation between distance and redshift of cosmological objects (Hubble’s law).

However there are some features that may seem very unlikely. The first is the isotropy of the CMB, which we today observe around 2.7 K on the whole sky. The natural thing to assume is that there is some physical process that smooth out any temperature variation that existed in the early universe. For this to happen, these regions must be connected by causal physics. The distance that causal physics can act is given by the particle horizon. Assume that we live in a spatially flat universe with dust and a cosmological constant. Then, when calculating the angular size of the particle horizon at last scattering as seen on the sky today, this only covers a few degrees. How is it then possible that regions on the sky today that are separated by as much as 180 degrees have almost the same temperature? We may assume that that the uniform temperature was a part of the initial conditions of the Big Bang model, but many may not be satisfied with this assumption. To explain this, cosmologists have postulated the existence of a process called inflation. Inflation is a short period of exponential growth in the early universe, i.e., the universe expands rapidly. Before inflation the regions visible to us were inside the particle horizon, thus there is no problem understanding the isotropy of the CMB.

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Inflation also solves other problems in the universe. Observations tell us that the curvature of the universe is close to flat, or equivalently that the energy density in the universe is close to the critical density. This is a problem because, as time goes by, the deviation of the density from the critical one increases. Since the density is close to the critical today, it would be much closer to the critical in the past. Again, we could assume that the universe did start in this fine tuned state, but we do not get an explanation why it is so. Inflation solves this problem because during the rapid expansion of the universe, any deviation from the critical will be wiped out.

Another puzzle with the observed universe is the magnetic monopole problem. In a very hot, early universe it is predicted that a number of heavy, stable particles, such as the magnetic monopole, would be produced. The problem is that we have not observed these particles. Inflation may solve this if the rapid expansion occurred after the production of monopoles. During expansion of space the density of these particles would decrease, hence becoming very rare in the present observable universe.

The idea of inflation was proposed by Guth (1981) [19], Linde (1982) [20] and Albrecht et al. (1982) [21] to solve the horizon, flatness and monopole problems. It was soon discovered that the same mechanism that explains the uniformity of the temperature in the CMB can also explain the origin of perturbations in the universe.

In its simplest form, the inflation is driven by a single scalar field. Quantum mechanics limits how homogeneous this inflationary field can be. The Heisenberg uncertainty principle for energy and time, ∆E∆t ≈ ~, sets a limit for how precise we can know the energy of the field in a given time interval. As a consequence of this, inflation will begin and end at different times in different regions of space and this leads to perturbations in the energy density. These perturbations will manifest themselves in the radiation as well as in the matter distribution we observe today. The simplest form of inflation theory is a single scalar field with with adiabatic fluctuations. This CMB anisotropies should then have an approximately scale-invariant spectral index of primordial fluctuations, n ' 1, and have produce density perturbations that are random-phase and have amplitudes with a Gaussian distribution.

2.2 Recombination/decoupling

In the early, hot universe matter and radiation were tightly coupled. The high energy radiation did not give protons and electrons the possibility to form a stable atom. Due to the high density of free electrons the photons are often scattered off baryons. This cause the mean free path of the photons to be short and they can not move far in the universe. This interaction between light and baryons coupled them in thermodynamic equilibrium, i.e., they shared the same temperature and they should have a blackbody spectrum.

The temperature drops as the universe expands. The binding energy of hydrogen is B_H= 13.6eV, and as the temperature drops such thatk_BT =B_Hone should expect the production of neutral hydrogen. But since there are many more photons than matter at this point, there is still enough high energy photons to ionize any hydrogen as the

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2.3 Acoustic oscillations in the baryon-photon fluid 21

temperature in the universe goes below 13.6 eV. As the temperature drops further free electrons can finally combine with hydrogen and helium nuclei. The formation of neutral atoms is referred to as recombination, despite the fact that electrons and nuclei never have been combined into atoms before. The loss of free electrons due the recombination and dilution from expansion decoupled the photons from the matter, and they could now move freely through the universe. This happened about 400 000 years after the Big Bang. We can see this radiation in all directions, and it forms a spherical shell around us with a radius of 13.7 billion light years. But this shell also has a thickness since the photons where released from the matter at different times.

The thickness is about 50 million light years. This radiation is the our best source to the early universe and we call it the cosmic microwave background. At the time of decoupling the temperature of the photons was about 3000 K. Since then the photons have lost their energy due to the expansion of the universe, and today we observe it with a temperature of 2.73 K.

But the CMB is not completely isotropic. There are temperature fluctuations of the order of 10⁻⁵. These have their origin from the quantum fluctuations in the scalar field responsible for inflation. These manifest themselves in the matter and photon densities, and then evolve with time. Because of the density differences at decoupling, the photons have different temperatures at different locations. The next section describe how the density varies with time after inflation has set up the original density perturbation and up to decoupling.

2.3 Acoustic oscillations in the baryon-photon fluid

Before recombination, the photons were coupled to protons and electrons, and they could be described as a single fluid. This fluid basically have two competing forces.

The gravity of the baryons makes the fluid clump together, resulting in over- and under-densities. The pressure from the photons, on the other hand, washes out these differences. As the over-density in the fluid grows, the pressure also grows. When the pressure from the photons gets large enough, it can withstand the contraction and cause it to stream out of the over-density. This makes the fluid being more homogeneous. If the pressure is large enough it may cause an under-density during this dissipation. But as the pressure is reduced, gravity will again cause the fluid to clump together.

These two forces will cause the fluid to oscillate. It oscillates at different scales at different times. The small scales starts first as they are first affected by causal physics.

The oscillations in the fluid are sound waves, and the sound speed in fluid,c_s, depends on the baryon density in the fluid. The sound speed determines how fast different parts in space can have causal contact with each other. The baryons makes the fluid heavier and reduces the sound speed. The maximum comoving distance traveled by a sound wave distance one can travel in the fluid by timeη is given by the sound horizon,

r_s(η) = Z _η

0

dη⁰c_s(η⁰).

whereη is the conformal time. The sound horizon is the distance a sound wave could

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propagate from the Big Bang to recombination, and is the largest scale on which causal physics can act in this fluid. On scales larger than this, we do not expect to see any fluctuations caused by acoustic oscillations. Instead, we only expect to see the effects that were caused by inflation. When decoupling occurs, the different scales are at different states in this oscillation. After decoupling the photons and baryons are no longer a single fluid, and the oscillation stops. This leaves us with various densities in space on different scales.

These various densities are the origin of the anisotropies in the CMB radiation.

In the remainder of this chapter I will look at the physical effects that caused the fluctuations in the Cosmic Microwave Backgound radiation. The first part looks at the changes to the CMB radiation that was sent out at decoupling. We may divide them into two groups, the primary and secondary fluctuations. The primary fluctuations were generated before the universe became transparent. The secondary fluctuations are the changes in frequency, apart from the change caused by universal expansion, while the radiation moved from the surface of last scattering and to us. The CMB radiation is basically a snapshot of the universe when it is around 400 000 years old.

This gives us a unique chance to observe the early universe. The anisotropies in the CMB remain small because the photons do not clump, so the distribution looks quite the same today as when the photons were sent out. There has only been linear changes to the photons.

There are also other sources causing microwave radiation other than the CMB.

These will contaminate the “pure” signal sent out. All of these effects are summarized in Table 2.1

2.4 Primary fluctuations

In this section I will look at the effects that cause anisotropies in the CMB during recombination.

2.4.1 Sachs-Wolfe effect

First is the Sachs-Wolfe effect. If the radiation has to climb out of the gravitational potential it is thereby redshifted. This has two effects: i) Photons loose energy as they move out of the gravitational field.

µ∆T T

¶

I

=−∆φ c² ,

where ∆φis the difference in gravitational potential at the emitter position and at the observer position; ii) Because of the gravitational time dilatation time proceeds at a slower rate down in the gravitational field. We then seem to be looking at a younger, and hence hotter, universe where there is an overdensity.

µ∆T T

¶

II

= 2

3(1 +ω)

∆φ c² ,

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2.4 Primary fluctuations 23

Primary Sachs-Wolfe (Gravity) Doppler

Density fluctuations Damping

Secondary

Gravity

Early ISW Late ISW Rees-Sciama Lensing Local reionization Thermal SZ

Kinematic SZ Global reionization

Suppression New Doppler Vishniac Other

Extragalaxtic Radio point sources

sources IR point sources

Galactic

Dust Free-Free Synchrotron Local

Solar system Atmosphere Noise, etc.

Table 2.1: Sources of temperature fluctuations in the CMB. The two topmost categories shows what effects that change the CMB directly. The last category shows physical processes that produce microwave radiation that contaminate the CMB signal.

Here ω is the equation of state factor that determine the relation between pressure p and matter/energy density ρ. For non-relativistic matter, also just called dust, this constant is equal to 0. In this case the effect becomes

µ∆T T

¶

II

= 2 3

∆φ c² , The total of this then gives µ

∆T T

¶

SW

=−1 3

∆φ c² .

2.4.2 Doppler effect

At last scattering, the fluid moves in a random direction with a velocity,v, relative to us. If the fluid in a direction ˆn moves towards us then the photons sent out will be blueshifted, while if the fluid moves away from us, the photons will be redshifted.

µ∆T T

¶

D

= v·n c .

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2.4.3 Adiabatic fluctuations

In a region with higher matter density the photon density will also be higher. For an adiabatic density fluctuation the photon density fluctuation is related to the matter density fluctuation as µ

∆ρ ρ

¶

γ

= 4 3

µ∆ρ ρ

¶

m

The density is proportional to temperature, n_γ ∝ T³, hence a region with increased photon density will have a higher temperature. Last scattering happens at roughly the same temperature. The temperature decrease as the universe expands, so we might expect that the fluctuations in photon temperature can not be observed as they all were sent out with the same temperature. But there will be an observable difference due to the fact that regions that originally had different temperature will send out the photons at different times. Overdense regions with higher temperature will scatter off photons at later times than the regions with a lower temperature. This will cause radiation from overdense regions to have a lower redshift from universal expansion. The temperature difference caused by this effect is

µ∆T T

¶

A

=− ∆z

1 +z = ∆ρ ρ ,

wherez is the redshift and the last equality assumes linear growth, ∆ρ∼(1 +z)⁻¹. Hence the total effect of these three become

µ∆T T

¶

tot

=−1 3

∆φ

c² +v·n c −∆ρ

ρ

2.4.4 Damping by photon diffusion

We treat the baryons and photons as a single fluid, but this is just an approximation as the photons will travel a finite length between scatters. This way photons can travel between the hot and cold regions, and any perturbation on scales smaller than this length can be expected to be washed out. This effect is named Silk damping after Joseph Silk who first discussed the effect in 1968 [22]. It is also known as diffusion damping.

2.5 Secondary fluctuations

Next we got the effects that altered the CMB radiation after decoupling when traveling to us today.

2.5.1 Change in gravitational potential

The first effect is caused by gravity. Even if the photons now are decoupled from the baryons, they are still affected by the gravitational potential of matter.

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2.5 Secondary fluctuations 25

This effect can be split into three parts, all due to time-variations in the gravitational potential. I.e., ˙φ6= 0 where ˙φdenotes the conformal time derivative of the gravitational potential. When a photon flies through a potential well it is blueshifted when falling in to it, and it is then redshifted when climbing out again. As long as the potential is the same on entrance and exit the net shift in frequency of the photon is zero. But if the potential changes while the photon travels through it, the frequency of the photon will change. During the matter-dominated era the potentials remains the same. But when other components dominate the universe these potentials may change.

At the time of last scattering the photon-contribution to the density of the universe is significant. This makes the potentials decay, causing the early ISW effect.

At recent times the universe seems to be dominated by vacuum energy which causes the universe’s expansion to accelerate. This vacuum energy also causes the potentials to vary with time, and the radiation’s energy will change. This is called the late ISW effect.

When non-linear structures (such as galaxy clusters) form, linear perturbation theory breaks down and the perturbation theory result ˙φ = 0 is no longer valid. This is called the Rees-Sciama effect.

The other effect caused by gravity change the direction of the photons and not the energy. If a pair of photons would have arrived separated by an angleθin the absence of fluctuations in φ, they will in reality arrive separated by some angle θ+ ∆θ. This change in the photons trajectory is called weak gravitational lensing. The total power of the in the fluctuations are conserved, but the power is redistributed from the peaks to the troughs.

2.5.2 The Sunyaev-Zel’dovich effect

The photons can be directly influenced by baryonic matter as they free-stream towards us. Baryons may be reionized after recombination locally e.g., in hot clusters of galaxies or globally throughout all of space.

Local reionization is called the Sunyaev-Zel’dovich (SZ) effect and is caused by two effects: i) If a cluster of galaxies is moving towards us, Thomson scattering of the CMB photons off free electrons in the hot gas in the galaxies will cause a Doppler blueshift in the direction of the cluster. This is known as the kinematic SZ effect. ii) The high energy of the free electrons will change the Planck spectrum by reducing the Rayleigh- Jeans (low ν) tail and increase the Wien (highν) tail. This happens independently of the clusters velocity and is known as the thermal SZ-effect.

2.5.3 Global reionization

If global reionization happens throughout space it will suppress fluctuations on small scales in the CMB power spectrum. This reionization causes the photons to Thomson scatter off free electrons, thus changing the original direction of the photon. Then we don not know exactly which direction the original photon came from. It could originally

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have come from anywhere within a certain region on the sky, and the signal is therefore smeared out at these scales.

2.6 Other sources to microwave radiation

There are of course other sources for radiation at microwave wavelength other than the radiation from the last scattering. This will contaminate the signal from the cosmic microwave background. All sources of microwave radiation that might come between us and the radiation left over from the Big Bang are called foregrounds. But the problems with foregrounds are manageable. The various effects causing the radiation is frequency dependent, and this is something we may take advantage of if we observe the sky in many different frequencies (see section 7.1).

2.6.1 Extragalactic radiation

Extragalactic point sources comes from outside our own galaxy. We distinguish between radio and infra-red sources. To get rid of this radiation we may subtract it from known point source catalogues or we may throw away the pixels containing a bright point source.

2.6.2 Galactic radiation

There are at least three sources for radiation from our own Galaxy: dust, free-free emission and synchrotron radiation. The amplitude of each component varies across the sky, but the relative amplitudes are quite typical. The effects from dust is caused by two processes, namely thermal radiation.and radiation from charged spinning dust grains.

Free-free emission (Bremsstrahlung) is produced when a charged particle, e.g., an electron, is passes near another charged particle, such as an atomic nucleus. The electron loses energy as it passes near the ion and the energy difference due to this acceleration is sent out as electromagnetic radiation. Synchrotron radiation is generated when ultrarelativistic, charged particles are accelerated through magnetic fields.

The amplitudes of the of these foregrounds have proven to be smaller than the cosmic signal in a wide part of the frequency range. At high frequencies, dust dom- inates the signal, and at low frequencies synchrotron and free-free radiation become dominant.But in the range from 30 to 200 GHz, the CMB signal often has the largest intensity.

2.6.3 Local radiation

Contamination may also come from our own solar system. Radiation can come from the sun, moon, earth and other planets. The instrument we are using to register the radiation may have some electronic receiver noise. Ground- and balloon-experiments

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2.6 Other sources to microwave radiation 27

will be affected by radiation in the atmosphere. There are of course many potential sources of systematic errors.

COBE is a real life example where local interference can false data. For about two months each year the satellite went behind the shadow of the Earth. This caused the signal to be non-Gaussian. It took quite some time to discover, but after removing the data from the two months when the Earth obscured the satellite, the non-Gaussian behaviour was gone [23].

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Chapter 3

Mathematical description of the CMB

The physics described in Chapter 2 comes from a thorough mathematical description of the evolution of the perturbations in the universe from the time of inflation to today. These can be described using the Boltzmann equations and the theory of general relativity. The deduction of the equations describing be perturbations as the universe evolve is a demanding task, and is outside the scope of this thesis. For great texts on this subject I recommend Dodelson [16] and Tegmark [17]. Instead, I will in this chapter go through how the we describe the CMB signal we are receiving and how we handle it in order to do calculations with them.

3.1 Data model

First we need a model of the observed signal. The signal d(ˆn) we receive from a direction ˆnon the sky can be written as

d(ˆn) =s(ˆn) +n(ˆn) +f(ˆn) (3.1) Heresis the clean signal from the CMB with power spectrumC_l,nis the instrumental noise andfis the sum of foreground contributions. The direction may also be written in polar coordinatesθandφ. Many instruments, like COBE and WMAP, do not measure the absolute temperature of the CMB radiation, but the temperature difference in two different directions. Hence the signal is given ass(ˆn) = ∆T(ˆn) =T(ˆn)−T₀, where T₀ is the mean temperature of the CMB radiation, 2.73 K.

However the instruments that register the CMB signal do not just observe a single point on the sky, but a finite solid angle. This smears out the signal and the small scales get filtered away;

d(ˆn) =A(s(ˆn) +f(ˆn)) +n(ˆn), (3.2) whereAdenotes convolution by an instrumental beam. This process can be described mathematically as a convolution in pixel space or a multiplication in harmonic space.

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There are several advantages working in harmonic space, so we will introduce spherical harmonics.

In normal flat space we can expand a function into wave-functions through a Fourier transform. This makes it possible to decompose the function into different scales, or modes. The function we are interested in is the CMB temperature field we observe on the sky. The desired wave-functions are given by Laplace’s equation ∆²ψ= 0. As we observe the temperature field on a sphere, we must solve this equation in spherical coordinates. The solution to this equation is

ψ≡Y_`m(θ, φ) = s

2`+ 1 4π

(`−m)!

(`+m)!P_`^m(cosθ)e^imθ

for ` ≥ 0 and −` ≤ m ≤ `. The Y_`m(θ, φ)’s are called spherical harmonics and are eigenfunctions of the angular part of the Laplacian. and the P_`^m’s are the Legendre polynomials.

The temperature field can be expanded in terms of the spherical harmonics

∆T(ˆn) = X∞

`=0

X`

m=−`

a_`mY_`m(ˆn) (3.3)

The`andmare conjugate to the real direction we are observing, ˆn. This decomposition is completely analogous to a Fourier decomposition in flat space. Thea_`ms contain all the information about the temperature field. These functions are orthogonal, with

normalization Z

dΩY_`m(ˆp)Y_`^∗0m⁰(ˆp) =δ_``⁰δ_mm⁰

Multiplying both sides of Equation 3.3 byY_`^∗0m⁰(ˆp) and integrating gives a_`m=

Z

dΩY_`m^∗ (ˆp) ∆T

The different modes are referred to as multipoles. For a value`, the typical scale for a spot for is θ ∼180^◦/`. For ` = 0,1,2,3 and 4 the modes are referred to as the monopole, dipole, quadrupole, octopole and hexadecapole. `= 0 picks up the constant part, ` = 1 picks up the linear part, ` = 2 picks up the quadratic part ` = 3 picks up the cubic part, etc. This way we decompose a temperature map into the different

`-modes that corresponds to different scales. This decomposition is shown in Figure 3.1, where all the different “`-maps” sums up to produce the temperature map.

With the convolution by the instrumental beam, the CMB signal now becomes A∆T(ˆn) =

`Xmax

`=0

X`

m=−`

b_`a_`mY_`m(ˆn), beam convolved map (3.4) Here,b_` is the Legendre transformation of the experimental beam. Since small scales are being smudged out, we can only observe the scales up to some`_max. Smaller scales are simply not resolved by the instrument. How to choose this value will be determined later.

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3.2 The power spectrum 31

Figure 3.1: Decomposing the temperature map using spherical harmonics. Each set of

`corresponds to a certain scale on the sky.

3.2 The power spectrum

We need a way to quantify and describe the observed data. If we got a set of data, what could it tell us? We need to be able to characterize the distribution so that theory can be compared to experiments. It is one thing is to look at a map of the distribution of matter or CMB, and another thing to see what these maps quantitatively can tell us about cosmological models. We are only interested in the amplitude of the fluctuations for the different scales. Under the assumption of statistical isotropy and homogeneity, the specific positions of the maximum or minimum are irrelevant.

Maybe the most important statistics for the CMB and the large-scale distribution of matter is the two-point correlation function, called the power spectrum in Fourier space. In the case with distribution of mass, the mean density of the galaxies is ¯n, and the inhomogeneities in the distribution at a position~xis given byδ(~x) = (n(~x)−¯n)/¯n.

If we work in Fourier space it is easier to separate large scales from small scales. The Fourier transform of the inhomogeneity is ˜δ(~k), wherekdenotes the scales. The power spectrumP(k) is defined via the two-point correlation function

hδ(~k)˜˜ δ(~k⁰)i= (2π)³P(k)δ³(~k−~k⁰) (3.5) The angular brackets denote an average over the whole distribution andδ³() is the Dirac delta function which constrains~k =~k⁰. The power spectrum tells us about the spread, or variance, in the distribution. If there is a lot of over- and under-dense regions, then the power spectrum will be large. If the amplitude is small, then the distribution of matter is smooth.

The two-point correlation function is also the best measure of the anisotropies in the CMB. Instead of Fourier transforming the CMB temperature, it is better to expand it in spherical harmonics, since the temperature is a two-dimensional field defined on the sky. The temperature field expanded in spherical harmonics is given in Equation 3.3.

The a_`ms are drawn from a distribution which traces its origin to the quantum fluctuations first laid down during inflation. This distribution is said to be Gaussian.

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The mean value for the a_`ms are zero, ha_`mi = 0, but their variance is not zero.

Thea_`ms are Gaussian random variables, and if there is statistical isotropy, then they are statistically independent for different ` and m. We can then write ha_`ma^∗_`0m⁰i = δ_``⁰δ_mm⁰C_` where C_` is the variance of the a_`ms. It is important to note that for each

`thea_`ms have the same variance, hence we only write the variance asC_`.

For each` we get (2`+ 1) differenta_`ms. For low values of` we are only sampling a few modes. For ` = 2 we only measure five components, and we do not get much information about the variance forC₂. Therefore is a fundamental uncertainty in the knowledge we may get about theC_`s. This uncertainty is called the cosmic variance.

µ∆C_` C_`

¶

cosmic variance

= r 2

2`+ 1 (3.6)

This is most pronounced at low `, while for higher values of ` we can sample more modes and the variance decreases. This effect is indicated by the blue band present in Figure 3.2.

We are not interested in the details in the observed CMB, but in its statistical properties. The quantum fluctuations from inflation were random, so our universe is just one realization of a stochastic ensemble of universes. Just as we know the statistical properties of throwing two dice, we will get different outcomes in different series of throwing them. So although the outcomes looks different, they have the same statistical properties.

The angular power spectrum (or anisotropy spectrum) is defined by, C_` = 1

2`+ 1 X`

m=−`

a_`ma^∗_`m=

|a_`m|²®

(3.7) Here l is the multipole number and is related to the angular extension on the sky. It is given byθ'π/`radians = 180^◦/`. This tells us how strong variations there are on various scales. This is a very important function for cosmologists because the cosmological parameters alters the shape of the power spectrum. The parameters change the location, height and width of the peaks in different ways. So by measuring the power spectrum accurately, we can estimate the values of the different parameters. Figure 3.2 shows the power spectrum from the WMAP five year data.

3.3 Likelihood

Inflation theory predicts that the signal we observe on the sky is drawn from a Gaussian distribution. The probability distribution for a multivariate normal distribution is

p(x) = µ 1

2π

¶ⁿ

2 1

p|C|e⁻¹²^(x−µ)^T^C⁻¹^(x−µ) (3.8) Here,xis the data from the observed sky consisting ofnelements andCis the covariance matrix. We call this distribution the likelihood and from now write it asL(x). We

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3.3 Likelihood 33

Figure 3.2: The temperature power spectrum with points from WMAP five year data.

The full line is the best fit for the ΛCDM model and the shaded region around the curve shows the uncertainty due to cosmic variance. The horizontal axes show how the angles on the sky (upper axis) corresponds to the multipoles`(lower). Courtesy of the WMAP science team.

will study the centered distribution, i.e., µ= 0. Also, it is standard practice to work with the logarithm of the likelihood, as this gives more manageable numbers. Using thousand pixels gives us a log likelihood of the order of 1000, while the likelihood is e¹⁰⁰⁰. The large number in natural units may cause some numerical errors, hence it is better to work with the logarithm of the likelihood. Taking the logarithm of the above expression gives

−2 lnL(x) =nln 2π+ ln|C|+x^TC⁻¹x (3.9)

In this analysis I will be dropping the first term, as it is just a normalizing constant, and it does not change where the parameters gives the highest likelihood.

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3.4 Signal covariance matrix

To calculate the likelihood we must first find the covariance matrix for the data. For a vector

X=



 X₁

... X_n





the covariance matrix is defined as

C_ij =cov(X_i, X_j) (3.10)

=E[(X_i−µ_i)(X_j−µ_j)] (3.11)

=h(X_i− hX_ii)(X_j − hX_ji)i (3.12) The last equality is just a different type of nomenclature. BothE[] andhidenotes the expectation value of the given element.

Adapting the data model in Equation 3.1 the covariance matrix of the observed sig- naldis given byC=h(d_i−hd_ii)(d_j−hd_ji)i=h(d− hdi)¡

d^t− hdi^t¢

i=h(s+n+f− hs+n+fi)

¡(s+n+f)^t− hs+n+fi^t¢

i. The expectation value of the three components are zero, hsi = hni = hfi = 0. We assume that none of the three components are internally correlated, so the cross terms all are zero, hsn^ti =hsf^ti =hnf^ti = 0. The covariance matrix becomes

C=hss^ti+hnn^ti+hff^ti ≡S+N+F (3.13) HereSis the covariance matrix of the clean CMB radiation,Nis the covariance matrix of the noise from the instrument, andFis the covariance matrix of the foregrounds.

We utilize the fact that we are observing the signal on a sphere and expand the signal in terms of spherical harmonics, as described in Equation 3.3. Inserting the expression for the spherical harmonics into the CMB covariance matrix gives us

S_ij =hs( ˆn₁)s^∗( ˆn₂)i=

* _∞ X

`=0

X`

m=−`

a_`mY_`m( ˆn₁) X∞

`⁰=0

`⁰

X

m⁰=−`⁰

a^∗_`0m⁰Y_`^∗0m⁰( ˆn₂) +

The spherical harmonics are just constants and we may pull them out of the averaging brackets,

S_ij = X∞

`=0

X`

m=−`

X∞

`⁰=0

`⁰

X

m⁰=−`⁰

Y_`m( ˆn₁)Y_`^∗0m⁰( ˆn₂)ha_`ma^∗_`0m⁰i

We assume that the CMB field is Gaussian and isotropic, hence we do not depend on the actual directions. This means that`=`⁰ and m=m⁰, orha_`ma^∗_`0m⁰i=δ_``⁰δ_mm⁰C_`. The Legendre polynomial P_` may be written in terms of a sum of products of the spherical harmonics P_`(ˆx·xˆ⁰) = _2`+1^4π P_`

m=−lY_`m(ˆx)Y_`m(ˆx⁰), and the signal covariance matrix therefore reads,

S_ij = X∞

`=0

X`

m=−l

Y_`mY_`m^∗ C_`= 1 4π

X∞

l=0

(2`+ 1)C_`P_`(cosθ_ij), (3.14)

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3.5 Technical considerations 35

where ˆn₁·nˆ₁= cosθ_ij. Here we choose to parameterize the power spectrum with a free amplitudeqand tiltn like

C_` =q µ`

`₀

¶_n

C_`^fid, (3.15)

whereC_`^fid is a fiducial model and `₀ is a pivot multipole. For a beam convolved map, the signal covariance matrix becomes

S_ij = 1 4π

`Xmax

l=0

(2`+ 1) (b_`p_`)²C_`P_`(cosθ_ij), (3.16) where p_` is the effect of finite pixelization and principally acts the same way as the beam functionb_`. Due to the beams finite resolution, the instrument can only resolve scales up to a multipole `_max. The value of this will be discussed in Section 3.5.1.

We assume that the noise is Gaussian and uncorrelated from pixel to pixel, with a standard-deviation ofσ_p. The covariance matrix then becomes

N_ij =hn_in_ji=δ_ijσ²_i, (3.17) whereiand j are two pixel indices. Thus the noise covariance matrix is diagonal, with variances on the diagonal.

We also need a term that could remove unwanted effects. In practice this is done by adding an extra term in the covariance matrix. If we want to be insensitive to a signalf(ˆn) then we want it to have zero statistical weight. To do this we say that the uncertainty of this component is “infinitely” large. In practice it is done by adding the following term to the covariance matrix,

F=λff^t (3.18)

Here,λis a large constant and f is a signal on the sky, also called template, we wish to be insensitive to. ff^t is the outer product of the template. More detailed information on the various signals will given in the WMAP-data section 7.1.

3.5 Technical considerations

3.5.1 Determining resolution parameters

There are two things we must think about when choosing`_maxand the pixelization to represent the observed data. First of all we know that the resolution of the instrument beam determines what scales are measurable. `_max must therefore be chosen large enough so that there is only negligible power beyond this multipole.

A Gaussian beam is a good approximation to the instruments that carry out the CMB experiments. The total width of the beam profile extends forever, so as a measure of the beam size, the Full Width Half Maximum (FWHM),θ_FWHM, of the beam is used.

As the name suggests, FWHM is the width of the beam pattern where the beam drops

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0 100 200 300

Multipole, l

0 0.2 0.4 0.6 0.8 1

B(l)

Figure 3.3: Show three different Gaussian beams. The full line shows a beam with θ_FWHM = 2.20 which corresponds to a resolution supported by N_side = 64. Using this beam, the typical scales observed is`_FWHM∼80, but it registers signal for scales much smaller than this. We can see that we can choose `_max considerably larger, around 130 for `_max = 2N_side or 190 for `_max = 3N_side. The dashed line is a beam with θ_FWHM = 4.50 and is supported by resolution at N_side = 32. The dotted line shows a beam withθ_FWHM= 9.00 which corresponds to a resolution supported by N_side= 16.

to half of its maximum. This gives us a number on the scales that are resolved by the instrument.

The angular extension on the sky is related to the multipole number as θ ' π/`.

So for the FWHM, the typical scales the beam can resolve is`_FWHM 'π/θ_FWHM. But the FWHM is just a measure of the typical scales for the beam, and the beam does resolve scales that are smaller than this. The Fourier transform of the beam profile is also a Gaussian, given as,

B(`) =e^−`(`+1)σ²^/2 (3.19) where σ = θ_FWHM/(p

8 ln(2)). Using this formula, I have plotted five different beam functions in Figure 3.3. Here we can see how the beam falls off with`. For high enough

`the beam suppresses the signal, and can not register the fluctuations on these small scales. Or put another way, the beam can not resolve these small fluctuations, and they are just smeared out. This gives us the largest multipole `_max. We must choose `_max such that there is negligible power beyond this multipole. For the case of the smallest beam, indicated in red, the typical scales are`_FWHM ∼80. We can see that the beam does have much power above this multipole, and has almost gone down to zero around

`∼200. This may seem like a good value for choosing `_max.

But there is a second aspect we must think about. We can not choose`_maxto be too

Bayesian Analysis of Hemispherical Power Asymmetry in the Cosmic Microwave Background